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Hawking Temperature for Near-Equilibrium Black Holes

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Hawking Temperature for Near-Equilibrium Black Holes KUNS-2372 Hawking temperature for near-equilibrium black holes Shunichiro Kinoshita1, ∗ and Norihiro Tanahashi2, † 1 Department of Physics, Kyoto University, Kitashirakawa Oiwake-Cho, 606-8502 Kyoto, Japan 2 Department of Physics, University of California, Davis, California 95616, USA (Dated: June 18, 2018) We discuss the Hawking temperature of near-equilibrium black holes using a semiclassical analysis. We introduce a useful expansion method for slowly evolving spacetime, and evaluate the Bogoliubov coefficients using the saddle point approximation. For a spacetime whose evolution is sufficiently slow, such as a black hole with slowly changing mass, we find that the temperature is determined by the surface gravity of the past horizon. As an example of applications of these results, we study the Hawking temperature of black holes with null shell accretion in asymptotically flat space and the AdS–Vaidya spacetime. We discuss implications of our results in the context of the AdS/CFT correspondence. PACS numbers: 04.50.Gh, 04.62.+v, 04.70.Dy, 11.25.Tq I. INTRODUCTION The fact that black holes possess thermodynamic properties has been intriguing in gravitational and quantum theories, and is still attracting interest. Nowadays it is well-known that black holes will emit thermal radiation with Hawking temperature proportional to the surface gravity of the event horizon [1]. This result plays a significant role in black hole thermodynamics. Moreover, the AdS/CFT correspondence [2] opened up new insights about thermodynamic properties of black holes. In this context we expect that thermodynamic properties of conformal field theory (CFT) matter on the boundary would respect those of black holes in the bulk. In particular, stationary black holes correspond to thermal equilibrium of the dual field theory at finite temperature equal to the bulk Hawking temperature [3]. It is interesting to study if such thermal properties persist when time dependence is turned on. From a technical viewpoint, the original derivation of the Hawking radiation [1] was performed on a static or stationary background (precisely speaking, as an approximation at late time), and it is not obvious how to accommodate dynamical spacetimes into the scheme. From a physical viewpoint, such a study can be useful since realistic black holes in our universe are dynamical or at least quasistationary. For example, processes like black hole formation after a gravitational collapse or black hole evaporation due to the Hawking radiation may involve highly time-dependent phases, and it is not obvious how the Hawking radiation will behave in such systems. When time dependence is sufficiently weak, however, we may expect thermodynamic properties of a black hole to persist. This expectation partially stems from physical insight on ordinary thermodynamics systems, in which thermodynamic properties persists when the system is sufficiently near equilibrium and quasistationary. If black hole thermodynamics is robust enough, such insight on usual thermodynamic systems leads to the above expectation on dynamical spacetimes and black holes. Many studies have been made on generalizing Hawking temperature for dynamical black hole approaches [4–10], and they support such an expectation. arXiv:1111.2684v2 [hep-th] 27 Jan 2012 Within the context of the AdS/CFT correspondence, thermodynamic properties of dynamical black holes lead to new insights into the dynamics of strongly coupled field theory on the boundary. One typical problem in this line is thermalization processes in the boundary field theory, which is holographically modeled by formation of the bulk black hole and its equilibration into a stationary state. Much effort is devoted to such problems in connection with, for example, application of the AdS/CFT correspondence to QCD physics like that in the RHIC experiment, or to nonequilibrium phenomena in condensed matter physics and fluid dynamics [11–23]. In this context, the Hawking radiation in the bulk is interpreted as quantum fluctuation in the boundary theory, and it plays an important role in certain setups [24, 25]. Based on these interests, in this paper we consider nonstationary spacetimes, and try to determine the Hawking temperature measured by observers in the asymptotic region. One naive way to do this would be to define a certain time coordinate to associate temperature of the black hole (future) horizon to that measured by asymptotic observers. Even ∗Electronic address: kinoshita˙at˙tap.scphys.kyoto-u.ac.jp †Electronic address: tanahashi˙at˙ms.physics.ucdavis.edu 2 though this procedure is seemingly natural from the viewpoint of black hole thermodynamics or horizon dynamics, it is puzzling from the viewpoint of causality, since this temperature is determined using information in the future region for that observer. In the AdS/CFT setup, if we were to define temperature at a point on the boundary in this way, we would need information in the bulk region, which is not causally accessible from that boundary point. Such a property would not be desirable if we care about causality in the bulk and boundary, especially when time dependence comes into the story. To avoid such problems, we consider the conventional derivation of the Hawking radiation based on Bogoliubov transformations for nonstationary background. An advantage of this method is that we can naturally determine temperature measured by asymptotic observers using only information causally available to them.1 Particularly, we focus our attention on an eternal black hole rather than black hole formation by gravitational collapse because we are interested in near-equilibrium system such as transition from an equilibrium state to another equilibrium state.2 In other words, we focus on late-time dynamics after the black hole has formed. We clarify how the Hawking temperature changes when a black hole becomes dynamical, for instance, due to mass accretion to the black hole, and show that the Hawking temperature of dynamical black holes can be naturally associated with “surface gravity” of the past horizon. The paper is organized as follows. In Sec. II, we evaluate the Bogoliubov coefficients using the saddle point approximation and show that the “surface gravity” of the past horizon gives the Hawking temperature observed in the asymptotic region. In Sec. III, as illustrations of the use of our method, we consider applications to simple examples in asymptotically flat and AdS cases. We summarize and discuss implications of the results in Sec. IV. II. ESTIMATION OF BOGOLIUBOV COEFFICIENTS We estimate Bogoliubov coefficients in order to define Hawking temperature for a nonstationary background in this section. We consider an eternal black hole in thermal equilibrium with its surroundings, that is, in the Hartle–Hawking state. In asymptotically flat cases, we should immerse the black hole into a thermal bath to realize this state. In asymptotically AdS cases, on the other hand, it is naturally realized due to its boundary condition [26, 27] because the black hole is enclosed in the AdS boundary. We exploit the saddle point approximation to evaluate the Bogoliubov coefficients and determine the temperature from them. After reviewing usage of the approximation for the static spacetime in Sec. II A, we consider its extension to nonstationary spacetime in Sec. II B. In this extension to nonstationary spacetime, we introduce a quantity, κ(u), which may be interpreted as an extension of the surface gravity of static Killing horizon. In Sec. II C, we reinterpret this quantity from a geometric point of view, and clarify that κ(u) is naturally associated with the past horizon of the eternal black hole. A. Static spacetime and the saddle point approximation For simplicity, we consider the two-dimensional part of the spacetime consisting of time and radial directions, that is, we focus on the s-wave sector of radiation. We also adopt the geometric optics approximation, and neglect the backscattering of the waves due to the curvature of spacetime. We introduce the null coordinate u which gives a natural time for observers in the asymptotic region (null infinity). If the spacetime is stationary, this time corresponds to the Killing time. We introduce another null coordinate U which is the affine parameter on the past horizon. For the stationary case it becomes the familiar Kruskal coordinate. Because lines described by u = const. are outgoing null geodesics, geodesic equations give a relation U = U(u) between the two null coordinates. Now, we consider a massless scalar field.3 In the current setup, solutions of the field equation are simply given by arbitrary functions of each null coordinate. In a standard manner we can define positive frequency modes with −iωu respect to u and U, respectively. Then, the Bogoliubov transformation between two sets of modes uω e and u¯ e−iωUˆ is determined by the Bogoliubov coefficients ∝ ωˆ ∝ ∞ αωωˆ i ωˆ dU = du e±iωu−iωUˆ (u), (1) β ±2π ω du ωωˆ r Z−∞ 1 The essential part of the calculation is similar to Ref. [9]. 2 The derivations are quite similar even on the background spacetime with gravitational collapse and black hole formation. See also comments in Sec. IV. 3 In asymptotically AdS cases we should consider a conformally coupled scalar field. 3 where the upper and lower signs correspond to αωωˆ and βωωˆ , respectively. Now, to evaluate those coefficients by the saddle point approximation, we will consider the integral ∞ exp φ(u)du, (2) Z−∞ where dU φ(u) log iωu iωUˆ (u). (3) ≡ du ± − ′ Saddle points are located at u = u∗ given by φ (u∗) = 0, where d dU dU φ′(u)= log iω iωˆ , (4) du du ± − du and a prime denotes a derivative. Consider a static black hole, for example. When the surface gravity at the Killing horizon is κ, the relation between the two coordinates u and U is given by U(u)= exp( κu), which is the coordinate transformation for the Kruskal coordinate. Then, we have − − ′ −κu∗ φ (u∗)= κ iω iωκeˆ =0, (5) − ± − and the saddle point ∓iθ −1 ire 1 r i π u∗ = κ log = log θ , (6) − ωˆ −κ ωˆ − κ 2 ∓ ∓iθ where re 1 iω/κ.
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